Wave Relation Cross-Scans

Echo Round-Trip Distance Three-Row Scan

halve the clocked path

Three echo rows multiply speed by echo time, then halve the round trip to get one-way distance.

Example

Echo time measures out-and-back travel, so one-way distance is half the speed-times-time round trip. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Echo time is a round trip

An echo clock measures out-and-back travel time. Multiply speed by time to get the round trip, then halve it for the one-way distance.

d=vt2d=\frac{v t}{2}

Short echo time

An echo time of 2 s means round trip 8 m and one-way distance 4 m.

round trip=vt=4 m/s ⋅ 2 s=8 m,d=8 m2=4 m\text{round trip}=vt=4\ \text{m}/\text{s}\,\cdot\,2\ \text{s}=8\ \text{m},\quad d=\frac{8\ \text{m}}{2}=4\ \text{m}
Echo distance rowA pulse travels to a wall and the echo returns to the source.outbacksourcedwall

Middle echo time

An echo time of 4 s means round trip 16 m and one-way distance 8 m.

round trip=vt=4 m/s ⋅ 4 s=16 m,d=16 m2=8 m\text{round trip}=vt=4\ \text{m}/\text{s}\,\cdot\,4\ \text{s}=16\ \text{m},\quad d=\frac{16\ \text{m}}{2}=8\ \text{m}
Echo distance rowA pulse travels to a wall and the echo returns to the source.outbacksourcedwall

Long echo time

An echo time of 6 s means round trip 24 m and one-way distance 12 m.

round trip=vt=4 m/s ⋅ 6 s=24 m,d=24 m2=12 m\text{round trip}=vt=4\ \text{m}/\text{s}\,\cdot\,6\ \text{s}=24\ \text{m},\quad d=\frac{24\ \text{m}}{2}=12\ \text{m}
Echo distance rowA pulse travels to a wall and the echo returns to the source.outbacksourcedwall

Halving the round trip gives the distance

The round-trip column is speed times time. The final distance column is half of that because the pulse travels out and back.

vtvtd4 m/s2 s8 m4 m4 m/s4 s16 m8 m4 m/s6 s24 m12 m\begin{array}{c|c|c|c}v & t & v t & d \\ \hline 4\ \text{m}/\text{s} & 2\ \text{s} & 8\ \text{m} & 4\ \text{m} \\ 4\ \text{m}/\text{s} & 4\ \text{s} & 16\ \text{m} & 8\ \text{m} \\ 4\ \text{m}/\text{s} & 6\ \text{s} & 24\ \text{m} & 12\ \text{m}\end{array}
waves At 4 m/s, echo times 2 s, 4 s, and 6 s have round trips 8 m, 16 m, and 24 m, so one-way distances are 4 m, 8 m, and 12 m.